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On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by…
It has been assumed that it is possible to approximate the interactions of quantized BPS solitons by quantising a dynamical system induced on a moduli space of soliton parameters. General properties of the reduction of quantum systems by a…
We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the…
We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver's metric derivations. The definition hereby given is shown to be equivalent to many…
In this paper, we study a part of approximation theory that presents the conditions under which a closed set in a normed linear space is proximinal or Chebyshev.
We discuss the notions of circumradius, inradius, diameter, and minimum width in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space…
In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from…
Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…
Let $V$ be a valuation domain of rank one with quotient field $K$. We study the set of extensions of $V$ to the field of rational functions $K(X)$ induced by pseudo-convergent sequences of $K$ from a topological point of view, endowing this…
We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is…
We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally…
We characterise rectifiable subsets of a complete metric space $X$ in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ with $\mathcal{H}^n(E)<\infty$…
We study uniform $\epsilon-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness,…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
The Barankin bound is generalized to the vector case in the mean square error sense. Necessary and sufficient conditions are obtained to achieve the lower bound. To obtain the result, a simple finite dimensional real vector valued…
We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…
This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the…
The machine learning community has recently put effort into quantized or low-precision arithmetics to scale large models. This paper proposes performing probabilistic inference in the quantized, discrete parameter space created by these…